Claim: “The person making the positive claim always has the Burden of Proof!”
Rating: Mixed. (It’s not that simple.)
First, there are always many different types of Burden of Proof (BoP) at play at any given time during any discussion, or even while you are quietly pondering your thoughts in solitude. There are epistemic BoP’s which make your position rational, with your position being believing something to be true, disbelieving it (to believe false), or suspending judgement on it and holding no position either way. Regardless if you believe p or do not believe p, you still always have a BoP to hold that position as rational. Since believing p and not-believing p is a strict dichotomy, it is possible that since you have to be in either one set or the other, neither are able to be rationally held such as in the case of properly basic beliefs which are held arationally.
Second, there is a BoP if you are actually trying to convince someone or not called the Burden of Persuasion. If you are not trying to convince someone, such as you are merely relating your own personal doxastic states as some self-referential or autobiographical information to someone, or something is normatively or generally presumed true then you have no burden of proof to anyone else.
I tell you I believe that fairies do not exist. By believing that I am supporting the position, or claim, that ontologically no such beings as “fairies” exist. By expressing my belief pro tanto as to only be just that, an expression of my doxastic states I have an epistemic BoP to myself if I want that position to be rational, however I have no BoP to someone else to demonstrate, prove, or even give evidence as to why I believe that fairies do not exist. My relating of my belief states do not entail a burden of persuasion to convince you that fairies do not exist.
Similarly, if you refuse to believe fairies exist it isn’t dependent upon my beliefs whatsoever, but based upon your own evidential or epistemic reasoning for why you merely do not believe fairies exist (agnostic on p), or you in fact believe as I do that fairies do not exist. If I told you A=A is true, would you not believe me merely because I didn’t present any evidence for why I believe A=A is true?
Thirdly, there are times where things are so normative that no BoP attaches merely by making a claim. If I say “my name is Steve”, I have no onus to convince you that my name is Steve. If I say A=A is a fact of mathematics, I have no BoP to convince you that A=A, no onus to demonstrate A=A, no onus to prove A=A, no onus to even provide any evidence that A=A, and if you do not believe A=A or believe it false then you would have your own epistemic reasoning to why hold it false, and your own BoP if you wanted to convince anyone that A does not equal A. Good luck with that, as merely by believing A≠A you are in a state of irrationality.